CHEM. 542 Spring 2010
Taught by Dana D. Dlott and Nancy Makri
Course Outline
1.
Fundamentals of time-dependent quantum mechanics (NM ~4 lectures)
The time-dependent Schrdinger equation. Time-independent Hamiltonians and
stationary states.
Evolution of non
Chem. 542
Assignment by Nancy Makri
PROBLEM 9
Convert the Bloch equations for a TLS coupled to a harmonic dissipative bath to the site
(left/right) representation.
Chem. 542
Assignment by Nancy Makri
PROBLEM 8
(a) Calculate the equilibrium values of the TLS populations at a given temperature.
(b) Use the Redfield equations for a TLS coupled to a dissipative harmonic bath to
show that the population of the excited st
Chem. 542
Assignment by Nancy Makri
PROBLEM 7
Consider a two-state system, for which the eigenstates of the Hamiltonian are
1 =
where L , R
operator
1
( L + R ),
2
2 =
1
(LR
2
)
are orthonormal left- and right-localized states.
We define the density
= p1
Chem. 542
Assignment by Nancy Makri
PROBLEM 6
Consider a hydrogen atom, initially in its ground state. At t = 0 an electric field is turned
on in the z direction, whose strength changes linearly with time. What is the probability
that the electron is in t
Chem. 542
Assignment by Nancy Makri
PROBLEM 5
Consider a harmonic oscillator of frequency in its ground state. At t = 0 a time
independent perturbation H1 (t ) = x is turned on. Using first order time-dependent
perturbation theory, calculate as a function
Chem. 542
Assignment by Nancy Makri
PROBLEM 4
In class we calculated the propagator for a free particle of mass m moving in one dimension.
a)
Using the one-dimensional result, find the propagator for a free particle moving in three di mensions.
b)
Calcula
Chem. 542
Assignment by Nancy Makri
PROBLEM 3
(a)
For a time-independent Hamiltonian H , show that energy is conserved during time evolution.
(Hint: write the expectation value of the Hamiltonian at a time t and use the time evolution
operator to express
Chem. 542
Assignment by Nancy Makri
PROBLEM 2
In class we proved that time evolution preserves the norm of the wavefunction. Using again
the unitarity of the time evolution operator, show that wavefunction overlaps are also preserved,
i.e.,
2 (t ) 1 (t )
Chem. 542
Assignment by Nancy Makri
PROBLEM 1
The survival amplitude for a system initially described by the state (0) is defined as
C (t ) = (t ) (0)
i.e., C (t ) is a measure of the overlap of the propagated state with the initial one. The survival
prob
CHEM. 542 Spring 2010
Required Textbook: Schatz and Ratner: Quantum Mechanics in Chemistry
Suggested Textbook: Feynmans Lectures in Physics, vol. 3
On reserve in the Chemistry Library:
Feynman lectures on physics, vol. III (paperback or hardbound)
M. D. F